The present invention relates to a method for computationally synthesizing supergratings in waveguides to give desired reflectance spectra, and more particularly the method relates to synthesizing supergratings using Fourier analysis.
Gratings are optical devices used to achieve wavelength-dependent characteristics by means of optical interference effects. These wavelength-dependent optical characteristics can, for instance, serve to reflect light of a specific wavelength while transmitting light at all other wavelengths. Such characteristics are useful in a wide range of situations, including the extraction of individual wavelength-channels in Wavelength Division Multiplexed (WDM) optical communication systems, or providing wavelength-specific feedback for tunable or multi-wavelength semiconductor lasers.
The term xe2x80x9cmulti-wavelength gratingxe2x80x9d generally refers to a grating that is capable of controlling optical characteristics at a number of wavelengths, such as a grating that reflects light at several select wavelengths (which may correspond to specific optical communication channels) while remaining transparent to light at other wavelengths. In some situations, however, there is a need to set the optical characteristics for a continuous range of wavelengths, rather than at specific wavelength values. This is the case when trying to compensate for the unevenness of optical gain profiles in laser cavities and optical amplifiers by means of a grating. This latter type of specification is usually difficult to meet with traditional grating technologies.
Gratings are usually implemented by modulating (varying) the effective index of refraction of a wave-guiding structure. The variation of refractive index along the length of the grating is often referred to as the xe2x80x9cindex profilexe2x80x9d of the grating. These changes in index of refraction cause incident light to be reflected: in the case of an abrupt interface between two index values, light incident directly on the interface is reflected according to the Fresnel reflection law:             E      r              E      i        =                    n        i            -              n                  i          +          1                                    n        i            +              n                  i          +          1                    
where Ei and Er are the incident and reflected electric fields at the interface, respectively, and ni and ni+1 are the refractive index values on either side of the interface, see FIG. 1. Although this reflection phenomenon is most striking for refractive index steps. It also occurs in a more complicated form with continuous changes in refractive index. Grating devices utilizing both types of reflection phenomena exist. A grating derives its wavelength-dependent character from optical interference effects. This phenomenon is illustrated in FIG. 2: incident light is reflected by each grating feature (step change in index of refraction) and interferes, either constructively or destructively, to generate a wavelength-dependent reflectance spectrum. At a certain wavelength, all the individually weak reflections add up constructively, leading to strong grating reflectance. At a different wavelength, however, the phase relation between the individual reflections is different and the beams may add up to produce little or no grating reflectance, transmitting most of the light.
Gratings may be xe2x80x9cwrittenxe2x80x9d into the optical wave-guide in a variety of different ways, depending primarily on the material used. Fiber or glass guides, for example, often make use of photorefractiveness, a property of specially prepared glasses that allows their refractive index to be varied by exposing them to high intensity light (typically in the ultraviolet). Semiconductor gratings, on the other hand, are usually implemented as surface-relief gratings by etching a grating pattern into the surface of the semiconductor guide (which may then be buried following subsequent deposition). Etching the surface of the waveguide does not affect its real refractive index as photoinscription does, but rather varies the guide""s effective index. Nevertheless, this difference does not affect the operation of the grating.
A simple and common grating device known as a Bragg Grating is illustrated in FIG. 3. The Bragg Grating consists of a periodic variation in refractive index and acts as a reflector for a single wavelength of light related to the periodicity (known as pitch, xcex9) of the index pattern. It is frequently used in both semiconductor systems and fiber-optic systems where it is known as a Fiber Bragg Grating. The Bragg Grating can actually reflect at several wavelengths, corresponding to overtones of its fundamental pitch, which satisfy the relation: xcex=2xcex9neff/N, where N is a positive integer (typically 1 for the design wavelength) and the average effective index neff is generally wavelength-dependent. However, these higher-order wavelengths tend to be at quite different spectral regions than the fundamental, thus not making the Bragg Grating useful as a multi-wavelength reflector. Moreover, these higher-order wavelengths cannot be tuned independently of one another.
Wavelength Division Multiplexing (WDM) is a technology where many communication channels are encoded into a single optical cable by utilizing different wavelengths of light. Gratings are often used to separate or process these channels. Older grating technologies can process one wavelength at a time, forcing devices intended to process multiple wavelengths to employ a cascade of single-wavelength gratings. This is not an attractive solution because, on top of the additional losses that each grating creates, even a single grating occupies a considerable amount of space by today""s standards of integration. It is thus desired to have a single device capable of processing several wavelengths in a space-efficient manner.
A similar situation occurs in the realm of semiconductor lasers. It is widely accepted that WDM technology would greatly benefit from lasers capable of generating light at several wavelengths. The output wavelength of semiconductor lasers is largely determined by the presence of xe2x80x9cfeedback elementsxe2x80x9d around or inside the laser gain section, which act to reflect light at the desired wavelength back into the laser. For multi-wavelength operation, multi-wavelength feedback is needed. Again, single-wavelength grating technology can only address this demand with a cascade of Bragg Gratings, leading to the same (if not more notable) loss and space problems mentioned above. The conclusion is the same: we desire a single device capable of generating multi-wavelength reflection and transmission spectra in a space- and resource-efficient manner.
There are several multi-wavelength grating technologies in existence: analog superimposed gratings, Sampled Gratings (SG), Super-Structure Gratings (SSG) and Binary Supergratings (BSG). The binary supergrating is also known as a binary superimposed grating, for historical reasons. BSG development was originally motivated by a desire to emulate the superposition of multiple conventional Bragg gratings, hence the term xe2x80x9cbinary superimposed gratingxe2x80x9d. Since then, synthesis techniques have evolved to allow the emulation of arbitrary diffraction characteristics, a flexibility better captured by the term xe2x80x9cbinary supergratingxe2x80x9d.
Analog superimposed gratings are a generalization of the Bragg Grating and are rooted in a principle of superposition: a grating profile consisting of the sum of the index profiles of single-wavelength gratings reflects at all of its constituent wavelengths. Such a grating relies on an analog index variation, that is, a refractive index that changes continuously along the grating length (FIG. 4). It is this analog nature of the profile that limits the functionality of these gratings: it is difficult to inscribe strong analog gratings using the photorefractive effect, as the change of index under illumination varies non-linearly with stronger exposures, making the writing process difficult. In semiconductors, where surface relief gratings are used, the situation is even worse as it is currently impossible to reproducibly etch analog features into the surface. The latter difficulty brought about the introduction of binary gratings, gratings that rely only on two refractive index values corresponding to the material being etched or not etched, illuminated or not illuminated.
Sampled gratings (SG) and superstructure gratings (SSG) represent two binary implementations of multi-wavelength gratings. The SG comprises alternating sections of grating and grating-free regions, and earns its name because it can be thought of as a grating which is sampled at specified intervals, see IEEE Photonics Technology Letters 5 489-491 (1993). This sampling produces diffraction spectra consisting of multiple reflectance peaks contained within a (typically) symmetric envelope. The SG is intrinsically limited in the flexibility in the location and relative strength of reflectance peaks, and, because of the large fraction of grating-free space, is also wasteful of real estate. The SG is therefore particularly unsuitable where a short grating is required or where waveguide losses are high.
With the super-structure grating (SSG), the grating period is chirped by finely varying the grating pitch, which corresponds to the length of one tooth-groove cycle. This can also be thought of as a sequence of finely-tuned phase shifts; common phase profiles include linear and quadratic chirp. Such an implementation in principle allows arbitrary peak positions and relative heights, but only at the expense of extremely high resolution, corresponding to a very small fraction of the size of the grating teeth themselves. For a typical semiconductor implementation, lithographic resolution of the order of 1 nm is required, see IEEE J. Quantum Electronics 29 1817-1823 (1993) and IEEE J. Quantum Electronics 32 443-441 (1996).
There are two main properties that define the binary supergrating in relation to other grating technologies The first is that the BSG relies on a discrete number of refractive index levels. This number is historically 2 and hence the BSG is known as a binary grating. Many of the advantages of the BSG, however, are maintained when multiple levels of refractive index are used, and most of the theory and methods presented here still apply. The second defining trait of the BSG is that it is a sampled structure characterized by a sample length. This refers to the fact that transitions between the grating""s index levels cannot occur at arbitrary positions, but, rather, must occur at multiples of the sample length. The BSG is strikingly similar in definition to the familiar notion of a digital signal, namely a discrete sampled waveform. Viewed as such, the BSG can be described by a series of (often binary) digits, indicating the refractive index setting at each sample point (see FIG. 5).
BSG design involves several key choices: selecting the refractive index levels for the device, as determined from material parameters and lithographic or photoinscription constraints; determining the desired sample length, considering the wavelength range important for the grating and the available lithographic resolution; setting a total device length for the grating, limited by the available physical space and the technological limitations of the inscribing process; and choosing the refractive index pattern of the sample-length sized segments that will produce the desired reflectance or transmittance characteristics.
One method of BSG synthesis is presented in Ivan A. Avrutsky, Dave S. Ellis, Alex Tager, Hanan Anis, and Jimmy M. Xu, xe2x80x9cDesign of widely tunable semiconductor lasers and the concept of Binary Superimposed Gratings (BSG""s),xe2x80x9d IEEE J. Quantum Electron., vol. 34, pp. 729-740, 1998.
This older method addresses the synthesis of xe2x80x9cmulti-peakxe2x80x9d gratings, gratings characterized by reflectance at several xe2x80x9cpeaksxe2x80x9d, which can be controlled in their position and strength. In this method, the designer begins with a set of sinusoids, each corresponding to a single reflectance peak and weighted according to that peak""s relative strength. These peaks are added together (i.e. superimposed; hence the BSG was initially known as a superimposed grating) to produce an xe2x80x9canalog profilexe2x80x9d. This profile is then quantized by a simple threshold method: if the analog profile value at a point is positive, let the corresponding BSG segment be of the high index value; if it is negative, let the corresponding BSG segment be of the low index value.
This method was sufficient to illustrate the BSG""s superiority in flexibility, efficiency and robustness when compared to prior multi-wavelength grating technologies, including the SG and SSG. However, it suffers from two deficiencies: firstly, the threshold quantization process introduces intermodulation distortion (see the DSM section), which largely limits the applicability of BSGs synthesized this way to active applications (laser feedback elements and the like). Secondly, this synthesis procedure is limited to multi-peak gratings, and offers little or no control over the individual peak shape. It is entirely incapable of generating flat-top channels, as desired by some communication applications, and of generating the near-arbitrary reflectance spectra demanded by some gain- and dispersion-compensation schemes. On a positive note, however, none of the prior (competing) grating technologies were capable of addressing these demands either,
In principle, there can be many methods for BSG synthesis. The simplest may be based entirely on trial-and-error. However, one quickly finds that these trial-and-error methods are most often computationally intractable. Instead, it is desirable to have a methodology that provides a more rigorous approach to methods of BSG synthesis.
The method disclosed herein provides an approach to BSG design based on the Fourier Approximation. The method divides the synthesis process into two stages: synthesis of an analog grating profile, followed by a quantization step. The method provides a generalized procedure for analog synthesis by drawing on the Fourier approximation and on FIR filter design theory.
In one aspect of the invention there is provided a method of designing a supergrating for a waveguide, comprising the steps of;
a) providing a reflectance spectrum in at least one spectral band to be produced by a supergrating in a waveguide, the reflectance spectrum having specified reflectance features, transforming said reflectance spectrum to a Fourier domain representation having Fourier-domain features;
b) computationally synthesizing an analog refractive index profile corresponding to the Fourier-domain representation; and
c) transforming said analog refractive index profile to a binary or multi-level refractive index profile representation in such a way as to conserve Fourier-domain information within said at least one spectral band.